Let $X$ be a curve (proper, smooth, ...) over a finite field $\mathbb F_q$ where $q$. Suppose also that $\mathbb F_q$ contains the $p$-th roots of unity, in this case we have the following (unique) chain of field extensions: $$\mathbb F_q \subset \mathbb F_{q^p} \subset \dots \subset \mathbb F_q^{(p)}$$ where each step in the extension is degree $p$ and the union of all these extensions is defined to be $\mathbb F_q^{(p)}$.
I am interested in the $\ell$ (possibly equal to $p$) torsion of the degree $0$ Picard group of $X_{\mathbb F_q^{(p)}}$. If we consider the algebraic closure $X_{\overline{\mathbb F_q}}$, then the $\ell$ torsion here looks like $P = (\mathbb Q_\ell/\mathbb Z_\ell)^{2\lambda}$ for $\lambda \leq g(X)$, the genus.
One strategy to get information about the Picard group of $X_{\mathbb F_q^{(p)}}$ would be to consider the Galois group $\Gamma = \mathrm{Gal}(\overline{\mathbb F_q}/\mathbb F_q^{(p)}) \cong \prod_{p' \neq \ell}\mathbb Z_{p'}$ and it's action on $P$.
Can this strategy prove the following:
1) If $\ell = p$, then the $\ell$-torsion in the Picard group over $\mathbb F_q^{(p)}$ is of the form $(\mathbb Q_\ell/\mathbb Z_\ell)^r$ for $r \leq 2\lambda$.
2) If $\ell \neq p$, then the $\ell$-torsion in the Picard group over $\mathbb F_q^{(p)}$ is finite.
Ideally, I would like to avoid any knowledge of the zeta function of curves.
